Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Nicola Fusco[1]; Virginia De Cicco; Micol Amar

  • [1] Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy;

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 3, page 456-477
  • ISSN: 1292-8119

Abstract

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New L 1 -lower semicontinuity and relaxation results for integral functionals defined in BV( Ω ) are proved, under a very weak dependence of the integrand with respect to the spatial variable x . More precisely, only the lower semicontinuity in the sense of the 1 -capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x . Under this further BV dependence, a representation formula for the relaxed functional is also obtained.

How to cite

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Fusco, Nicola, Cicco, Virginia De, and Amar, Micol. "Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 456-477. <http://eudml.org/doc/245653>.

@article{Fusco2008,
abstract = {New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega $) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.},
affiliation = {Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Complesso di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy;},
author = {Fusco, Nicola, Cicco, Virginia De, Amar, Micol},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {semicontinuity; relaxation; BV functions; capacity},
language = {eng},
number = {3},
pages = {456-477},
publisher = {EDP-Sciences},
title = {Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands},
url = {http://eudml.org/doc/245653},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Fusco, Nicola
AU - Cicco, Virginia De
AU - Amar, Micol
TI - Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2008
PB - EDP-Sciences
VL - 14
IS - 3
SP - 456
EP - 477
AB - New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega $) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.
LA - eng
KW - semicontinuity; relaxation; BV functions; capacity
UR - http://eudml.org/doc/245653
ER -

References

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